Question

if secθ - tanθ= p find value of secθ and tanθ

Answer 1

If $sec\theta - tan\theta = p$ find value of $sec\theta$ and $tan\theta$.

**Solution:**

Using a well-known trigonometric equation:

$$1 + tg^2\theta = sec^2\theta$$

we will have that

$$sec^2\theta - tg^2\theta = 1$$ $$(sec\theta - tg\theta)(sec\theta + tg\theta) = 1$$

from of the problem we know that

$$sec\theta - tan\theta = p$$

so

$$p * (sec\theta + tg\theta) = 1$$

where we can find that

$$sec\theta + tg\theta = \frac{1}{p}$$

So we have system of two equations

$$\left\{ \begin{array}{l} sec\theta - tg\theta = p \\ sec\theta + tg\theta = \frac{1}{p} \end{array} \right.$$

If we sum them we will find $sec\theta$, if we subtract from one second we will find $tg\theta$

$$\left\{ \begin{array}{l} 2sec\theta = p + \frac{1}{p} \\ -2tg\theta = p - \frac{1}{p} \end{array} \right.$$

So

$$\left\{ \begin{array}{l} sec\theta = \frac{1}{2}\left(p + \frac{1}{p}\right) \\ tg\theta = \frac{1}{2}\left(\frac{1}{p} - p\right) \end{array} \right.$$

Answer: $sec\theta = \frac{1}{2}\left(p + \frac{1}{p}\right)$ $tg\theta = \frac{1}{2}\left(\frac{1}{p} - p\right)$